| Version 3 |
Version 2 |
| \newcommand{\cD}[0]{\mathcal{D}} |
\newcommand{\cD}[0]{\mathcal{D}} |
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| Suppose $U$ is an open set in $\sR^n$ and $f$ is a locally |
Suppose $U$ is an open set in $\sR^n$ and $f$ is a locally |
| integrable function on $U$, i.e., $f\in L^1_{\scriptsize{\mbox{loc}}}(U)$. |
integrable function on $U$, i.e., $f\in L^1_{\scriptsize{\mbox{loc}}}(U)$. |
| Then the mapping |
Then the mapping |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| T_f: \cD(U) &\to& \sC \\ |
T_f: \cD(U) &\to& \sC \\ |
| u &\mapsto& \int_U f(x) u(x) dx |
u &\mapsto& \int_U f(x) u(x) dx |
| \end{eqnarray*} |
\end{eqnarray*} |
| is a zeroth order distribution \cite{hormander, lang}. (Here, $\cD(U)$ is |
is a zeroth order distribution \cite{hormander, lang}. (Here, $\cD(U)$ is |
| the set of smooth functions with compact support on $U$.) |
the set of smooth functions with compact support on $U$.) |
|
|
| \PMlinkname{(proof)}{T_fIsADistributionOfZerothOrder} |
\PMlinkname{(proof)}{T_fIsADistributionOfZerothOrder} |
|
|
| If $f$ and $g$ are both locally integrable functions on a open set $U$, |
If $f$ and $g$ are both locally integrable functions on a open set $U$, |
| and $T_f=T_g$, then it follows (see |
and $T_f=T_g$, then it follows (see |
| \PMlinkname{this page}{LocallyIntegrableFunction}), |
\PMlinkname{this page}{LocallyIntegrableFunction}), |
| that $f=g$ almost everywhere. Thus, the mapping $f\mapsto T_f$ |
that $f=g$ almost everywhere. Thus, the mapping $f\mapsto T_f$ |
| is a linear injection when $L^1_{\scriptsize{\mbox{loc}}}$ is equipped with |
is a linear injection when $L^1_{\scriptsize{\mbox{loc}}}$ is equipped with |
| the usual equivalence relation for an $L^p$-space. For this reason, |
the usual equivalence relation for an $L^p$-space. For this reason, |
| one also writes $f$ for the distribution $T_f$ \cite{lang}. |
one also writes $f$ for the distribution $T_f$ \cite{lang}. |
|
|
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{hormander} |
\bibitem{hormander} |
| L. H\"ormander, \emph{The Analysis of Linear Partial Differential Operators I, |
L. H\"ormander, \emph{The Analysis of Linear Partial Differential Operators I, |
| (Distribution theory and Fourier Analysis)}, 2nd ed, Springer-Verlag, 1990. |
(Distribution theory and Fourier Analysis)}, 2nd ed, Springer-Verlag, 1990. |
| \bibitem{lang} |
\bibitem{lang} |
| S. Lang, \emph{Analysis II}, |
S. Lang, \emph{Analysis II}, |
| Addison-Wesley Publishing Company Inc., 1969. |
Addison-Wesley Publishing Company Inc., 1969. |
| \end{thebibliography} |
\end{thebibliography} |
